Chemistry Reference
In-Depth Information
M
M
MM
−
γ
AA
AB
=
0
(2.32)
−
γ
AB
BB
or, equivalently,
1
2
2
2
γ
±
=
MM
+
±
(
MM
−
)
+
4
M
(2.33)
AA
BB
AA
BB
AB
The correct solution can be identified by taking the limit λ
B
= 0, that is, for pure A
we must have
1
2
(
γ=
MM M
AA
+
)
=
AA
(2.34)
AA
Therefore, we take the γ
+
solution of Equation 2.34 and equate it to unity to obtain
the implicit equation of state,
2
2
fTp
(,,
λ
AB
,
)
=
M
+
MMMM
+
(
+
)
+
4
−
2
=
0
(2.35)
AA
BB
AA
BB
AB
Taking the total differential of
f
, we have
∂
∂
f
T
∂
∂
f
p
dp
∑
∂
∂
f
df
=
dT
+
+
d
λ
=
0
(2.36)
i
λ
i
Using the Gibbs-Duhem (GD) expression (see Equation 1.11), we can derive all the
thermodynamic quantities of the system from the implicit equation of state Equation
2.35. As an example, the average density of A is
(
)
∂∂
f
fp
/
/
µ
∂
∂
p
A
pT
,,
µ
ρ
=
=−
(2.37)
B
A
(
)
µ
∂∂
A
p
,µµ
AB
,
T
,
µ
B
The entropy of the system is calculated from
∂
(
pV
T
)
∂
∂
p
T
(
∂∂∂
∂∂
fT
fp
/
/
)
p
p
,
µµ
µµ
,
S
=
=
V
=−
V
(2.38)
AB
∂
(
)
,
,
V
,
µµ
,
µµ
,
AB
AB
AB
A particular simple case is when
2
ΨΨΨ
AB
=
(2.39)
AA
BB
